Question

A 100kg mass traveling with a velocity of 15m/s on a horizontal surface strikes a spring with a spring constant k = 5 N/m a. Find the compression of the spring required to stop the mass if the surface is frictionless b. Find the compression of the spring if the surface is rough (0.4ku=0)

Answer 1

a.

By energy conservation:

KEi + SPEi + Wfr = KEf + SPEf

here, KEi = initial kinetic energy = 0.5*m*v^2

SPEi = initial energy stored in spring = 0

KEf = final kinetic energy = 0

SPEf = finally energy stored in spring = 0.5*k*x^2

Wfr = work done by friction = 0

given, m = mass = 100 kg

v = initial speed = 15 m/s

k = spring constant = 5 N/m

x = compression in spring = ??

then, 0.5*100*15^2 + 0 + 0 = 0.5*5*x^2

x = sqrt(20*15^2) = 67.08

x = 67.1 m

b.

Again, by energy conservation:

KEi + SPEi + Wfr = KEf + SPEf

here,

Wfr = work done by friction = Ff*d*cosA

Ff = friction force = k*N

N = normal force = m*g

given, m = mass = 100 kg

v = initial speed = 15 m/s

k = spring constant = 5 N/m

x = compression in spring = ??

k = 0.4

d = displacement of mass on rough surface = x

A = angle between friction force and displacement

then, 0.5*100*15^2 + (0.4*100*9.81)*x*cos(180 deg) + 0 = 0.5*5*x^2

2.5*x^2 + 392.4*x - 11250 = 0

by solving above quadratic,

x = 24.8 m